Final_1.pdf

theory (APT) and establish a link between APT and the common trends
model. This will be the focus of the discussion in the following section, on
common trends.

COMMON TRENDS MODEL AND APT

Earlier, in Chapter 2, the logarithm of stock price was modeled as a random
walk. To accommodate the common trends model, let us modify that a lit-
tle and consider the logarithm of the stock price to be a sum of a random
walk and a stationary series

log(pricet) = nt+et (6.9)

wherentis the random walk, and etis the stationary component. Differenc-
ing the logarithm of stock price yields the sequence of returns. Therefore,
based on Equation 6.9, the return rtat time tmay also be separated into two
parts

log(pricet) – log(pricet–1) = nt–nt–1+ (et–et–1) (6.10)

(6.11)

where is the return due to the nonstationary trend component, and is
the return due to the stationary component.
Notice that the return due to trend component is the same as the inno-
vation derived from the trend component. Therefore, the cointegration cri-
terion pertaining to the innovations of the common trend may be rephrased
as follows: If two stocks are cointegrated, the returns from their common
trends must be identical up to a scalar. But why in the world should stocks
ever have common returns? Is there a financial rationale for these to exist
among stocks? The answer to that is a resounding yes, and APT comes in
handy in providing a comprehensive explanation for this. Recalling the ear-
lier discussions on APT, stock returns may be separated into common factor
returns (returns based on the exposure of stocks to different risk factors) and
specific returns (returns specific to the stock). If two stocks share the same
risk factor exposure profile, then the common factor returns for both the
stocks must be the same. This provides us with a rationale for when we
might expect stocks to have a common return component.
We are now ready to draw parallels between the common trends model
and APT. According to APT, stock returns for a time period may be sepa-
rated into two types: common factor returns and specific returns. Let these
correspond to the common trend innovation and the first difference of the

rtc rts

rrrtt=+c ts

90 STATISTICAL ARBITRAGE PAIRS